A378981 -id:A378981 - cp's OEIS frontend

A378980 Numbers k such that (A003961(k)-2*k) divides (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

1, 2, 3, 4, 6, 7, 10, 25, 26, 28, 33, 46, 55, 57, 69, 91, 93, 496, 1034, 1054, 1558, 2211, 2626, 4825, 8128, 11222, 12046, 12639, 28225, 32043, 68727, 89575, 970225, 1392386, 2245557, 8550146, 12371554, 16322559, 22799825, 33550336, 48980427, 51326726, 55037217, 60406599, 68258725, 142901438, 325422273, 342534446

Offset: 1

Antti Karttunen, Dec 12 2024

nonn

Numbers k such that A252748(k) divides A286385(k).

Conjecture: Apart from a(5)=6, this is a subsequence of A319630, i.e., for all terms k<>6, gcd(k, A003961(k)) = 1. See also A372562, A372566.

Amiram Eldar, Table of n, a(n) for n = 1..69 (terms below 10^11; terms 1..60 from Antti Karttunen)
Index entries for sequences related to prime indices in the factorization of n.
Index entries for sequences related to sigma(n).

Cf. A000203, A003961, A252748, A286385, A319630, A372562, A372566.
Positions of 0's in A378981.
Subsequence of A263837.
Subsequences: A000396, A048674, A348514, A326134, A349753 (odd terms of this sequence).
Cf. also A378983.

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^e; q[k_] := Module[{fct = FactorInteger[k], m, s}, s = Times @@ f1 @@@ fct; m = Times @@ f2 @@@ fct; Divisible[m - s, m - 2*k]]; q[1] = True; Select[Range[10^5], q] (* Amiram Eldar, Dec 19 2024 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378981(n) = { my(u=A003961(n)); ((u-sigma(n))%((2*n)-u)); };
isA378980(n) = !A378981(n);

A379216 Difference 2*k - A003961(k) computed for k for which this difference divides difference (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, 1, 1, -1, -3, 3, -1, 1, 1, -43, 1, 5, 19, -1, -7, -5, 1, -2005, 1, -1, 149, -193, -1, -3, -79243, 1243, 1253, -7, 51, 581, -1, 3093, 1, 155491, 919, 1, -1, 15833, -877, -4295498497, 5129369, 31, 5779339, -69187, -29, 6745, 1, 181, 1, 69197, -397, -117433, -101, -1, 1, 2759, 1, -29479, 1, -5626288431709, 29669, -1, -132239, -1, -1, 14591, -2267959, -3187, 787250461

Offset: 1

Antti Karttunen, Dec 20 2024

sign

Among the initial 69 terms, there are eleven +1's and eleven -1's. The former correspond in A378980 with those of its terms that are in A048674 (1, 2, 3, 25, 26, 33, 93, 1034, ...), while the latter here correspond in A378980 with those of its terms that are in A348514 (4, 10, 57, 1054, 2626, ...).

Cf. A003961, A048674, A252748, A348514, A378980, A378981, A379217.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378981(n) = ((A003961(n)-sigma(n))%((2*n)-A003961(n)));
is_A378980(n) = !A378981(n);
for(n=1,2^25,if(is_A378980(n),print1((2*n)-A003961(n),", ")));

a(n) = -A252748(A378980(n)).

A379217 Quotient (A003961(k)-sigma(k)) / (2*k-A003961(k)) computed for those k for which this quotient is an integer, where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Original entry on oeis.org

0, 0, 1, -2, -1, 1, -3, 18, 9, -1, 17, 3, 1, -35, -7, -15, 57, -1, 339, -381, 3, -7, -969, -1213, -1, 3, 3, -979, 419, 29, -42735, 21, 731232, 3, 1445, 2809731, -4566981, 557, -19691, -1, 5, 544371, 5, -475, -1784691, 9051, 176870849, 808683, 280791301, 1803, -891775, -3679, -3733533, -444406677, 731480523, 275091

Offset: 1

Antti Karttunen, Dec 20 2024

sign

Terms in A378980 that correspond here with -1's are perfect numbers (A000396).

Antti Karttunen, Table of n, a(n) for n = 1..69 (based on b-file of A378980 computed by _Amiram Eldar_)
Index entries for sequences related to prime indices in the factorization of n.
Index entries for sequences related to sigma(n).

Cf. A000396, A003961, A252748, A378980, A378981, A379216.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378981(n) = ((A003961(n)-sigma(n))%((2*n)-A003961(n)));
is_A378980(n) = !A378981(n);
for(n=1,2^25,if(is_A378980(n),print1(((A003961(n)-sigma(n))/((2*n)-A003961(n))),", ")));

a(n) = A286385(A378980(n)) / A379216(n) = A286385(A378980(n)) / -A252748(A378980(n)).

A378982 a(n) = (A003961(n)-(1+sigma(n))) mod (A003961(n)-2*n), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 2, 0, 4, 0, 0, 16, 2, 3, 0, 0, 0, 35, 2, 20, 9, 2, 4, 74, 0, 0, 13, 42, 0, 32, 4, 0, 0, 2, 0, 133, 2, 1, 0, 98, 0, 68, 2, 3, 11, 4, 4, 280, 17, 6, 1, 5, 4, 254, 18, 176, 0, 2, 0, 146, 4, 1, 21, 0, 1, 50, 2, 9, 6, 86, 0, 479, 4, 8, 25, 11, 2, 86, 2, 380, 40, 2, 4, 270, 24, 8, 15, 170, 6, 290, 4, 15

Offset: 1

Antti Karttunen, Dec 13 2024

nonn

Cf. A000203, A003961, A252748, A286385, A378983 (positions of 0's).

Cf. also A378981.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378982(n) = ((A003961(n)-(sigma(n)+1))%((2*n)-A003961(n)));

a(n) = (A286385(n)-1) mod A252748(n).

cp's OEIS Frontend

A378980 Numbers k such that (A003961(k)-2*k) divides (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

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A379216 Difference 2*k - A003961(k) computed for k for which this difference divides difference (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

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A379217 Quotient (A003961(k)-sigma(k)) / (2*k-A003961(k)) computed for those k for which this quotient is an integer, where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

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A378982 a(n) = (A003961(n)-(1+sigma(n))) mod (A003961(n)-2*n), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

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